On the structure of 2-step nilpotent Lorentzian naturally reductive Lie groups

2026/07/20 by Brian Luporini, Silvio Reggiani, Francisco Vittone
#math.DG

paper · pdf

Abstract

We study 2-step nilpotent Lie groups with naturally reductive left-invariant Lorentzian metrics with respect to the presentation group N \rtimes H^aut. Replacing the standard non-degenerate center assumption with the weaker condition that the commutator ideal be non-degenerate, we develop a framework that extends the construction to the Lorentzian context and covers both the non-degenerate and degenerate center cases. In the degenerate case, we show that the associated Lie algebra is a central extension of a semidirect product whose Riemannian factor is naturally reductive. Furthermore, we obtain invariant decompositions of the defining representation, including a distinguished Lorentzian factor, and provide an explicit description of the isotropy algebra and the identity component of the isometric automorphism group. These results complete the structural description of naturally reductive 2-step Lorentzian nilpotent Lie groups under the assumption of non-degeneracy in the commutator ideal.

Related

• The distribution of symmetry of Lorentzian naturally reductive nilmanifolds
• Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups
• On Einstein Lorentzian nilpotent Lie groups
• Left-invariant Lorentzian flat metrics on Lie groups
• Ricci flat left invariant Lorentzian metrics on 2-step nilpotent Lie\n groups